A Geometric Approach to the Modified Milnor Problem

TL;DR

This paper links the modified Milnor Problem in group theory to a geometric conjecture about the fundamental groups of certain Riemannian manifolds, establishing equivalences and verifying cases for nilpotency and entropy gap phenomena.

Contribution

It demonstrates the equivalence between the modified Milnor Problem and the Nilpotency Conjecture in Riemannian geometry, and verifies the conjecture in specific cases.

Findings

Positive answer to the Milnor Problem (modified) is equivalent to the Nilpotency Conjecture.

Verified the Nilpotency Conjecture in some cases.

Established vanishing entropy gap phenomena for certain manifolds.

Abstract

The Milnor Problem (modified) in the theory of group growth asks whether any finite presented group of vanishing algebraic entropy has at most polynomial growth. We show that a positive answer to the Milnor Problem (modified) is equivalent to the Nilpotency Conjecture in Riemannian geometry: given $n, d>0$, there exists a constant $\epsilon(n,d)>0$ such that if a compact Riemannian $n$-manifold $M$ satisfies that Ricci curvature $\op{Ric}_M\ge -(n-1)$, diameter $d\ge \op{diam}(M)$ and volume entropy $h(M)<\epsilon(n,d)$, then the fundamental group $\pi_1(M)$ is virtually nilpotent. We will verify the Nilpotency Conjecture in some cases, and we will verify the vanishing gap phenomena for more cases i.e., if $h(M)<\epsilon(n,d)$, then $h(M)=0$.